(1) examples of infinitesimals
Several people have said that it's easy to give examples of
infinitesimals in nonstandard analysis. But nobody seems to have
commented on Harvey's highly relevant posting
FOM: 7:Undefinability/Nonstandard Models
on November 17. The essential point there is that it's impossible
to give an example of an infinitesimal under the Abraham Robinson
setup, because it's consistent with ZFC that there is no definable
nonprincipal ultrafilter on N, the natural numbers. If e were a
definable infinitesimal, then
{X subset of N: [1/e] is an element of X*}
would be a definable nonprincipal ultrafilter on N.
(2) f.o.m. testimonial
Lou and Dave have said that they personally find pure mathematics
more interesting than foundations of mathematics. I'd like to say
that I personally find foundations of mathematics incomparably more
interesting than pure mathematics. (I would also go farther and say
that foundations is *objectively* more interesting, not just more
interesting to me personally. But that is another story.)
First let me comment that my credentials in pure mathematics are
pretty good: (1) I went through the normal graduate training at MIT,
taking beginning and advanced graduate courses in many areas of pure
mathematics, including algebra, analysis, and topology/geometry.
(2) I had some very inspiring teachers in these areas, including
Guillemin and Quillen. (3) I subsequently made some serious
contributions to pure mathematics, perhaps most notably the
Carlson/Simpson dual Ramsey theory, which was used by Furstenburg in
his recent work in ergodic theory. The point I am trying to make is
that I had the option of becoming a pure mathematician, and I think
I could have done well in that direction, had I been so inclined.
Now to my main point. I opted not to pursue pure mathematics
because I found it too narrow, too confining, too technical (in the
sense that pure mathematicians place too much emphasis on technique
such as cohomology, and not enough on fundamental mathematical
issues). Instead I opted for foundations of mathematics. And by
the way, I always thought of the subject as "foundations of
mathematics," never as "mathematical logic" or "model theory" or
whatever. What drew me to foundations was its broad intellectual
appeal and scope, with its obvious philosophical significance and
the possibility of contributing to major intellectual developments
on a broad front. The work of Hilbert and G"odel and Friedman was
tremendously exciting. This is what moved me. Unfortunately,
because of circumstances (including a thesis advisor who was
explicitly hostile to foundations) it was quite a while before I was
able to pursue my foundational interests full time. But all along I
knew that foundations of mathematics was what I wanted to do, and
now I'm proud and happy to be doing it.
Sincerely,
-- Steve